топология / Farb, Margalit, A primer on mapping class groups

THE LANTERN RELATION VIA THE PUSH MAP

There is another way to derive the lantern relation that makes it much less mysterious. Let P be a pair of pants, that is, a sphere with 3 boundary components. Embed P in the plane and label the outer boundary component x and the inner components b1 and b2. We obtain an element of Mod(P ) by pushing b1 around b2, without ever turning b1 (think about a “do-si-do”).

From the Alexander Method and Figure 6.3 we see that this map is equal to

TxTb−1Tb−1.

1 2

More formally, this push map is an element of the image of the homomorphism π1(U T (A)) → Mod(P ), where A is the annulus obtained by capping b1 by a closed disk (see Section 5.2).

x

b2

b1

Figure 6.3 A push map.

Let S04 be a sphere with four boundary components. We have the following easy-to-see relation in π1(U T (P )) < Mod(S04), depicted in the left-hand side of Figure 6.4: pushing b2 around b3 and then pushing b2 around b1 is the same as pushing b2 around both b3 and b1. In other words, using the simple closed curves shown in the right-hand side of Figure 6.4, we have:

Since the Tbi are central in this group, we can rewrite this as

TxTyTz = Tb1 Tb2 Tb3 Tb4 .

And this is exactly the lantern relation.

Figure 6.4 A new view of the lantern relation.

FIRST HOMOLOGY OF THE MAPPING CLASS GROUP

It is a basic fact from algebraic topology that, for any path-connected space X, the group H1(X; Z) is isomorphic to the abelianization of π1(X). Since the homology of a group G is defined as the homology of any K(G, 1), we have that the first homology group of G with integer coefficients is

H1(G; Z) ≈ [G,GG] ≈ Gab,

where [G, G] is the commutator subgroup of G, and Gab is the abelianization of G.

THEOREM 6.2 For g ≥ 3, the group H1(Mod(Sg ), Z) is trivial. More generally, for any surface S with genus at least 3, we have that H1(PMod(S); Z) is trivial.

In other words, if the genus of S is at least 3, then the group PMod(S) is equal to its commutator subgroup, or, equivalently, PMod(S)ab is trivial. A group with this property is called perfect. As we will see below, the statement of Theorem 6.2 is false for g {1, 2}.

The following proof is due to Harer [71].

Proof. Let S be a surface whose genus is at least 3. Since Dehn twists about nonseparating simple closed curves are all conjugate (Fact 3.6) it follows that each of them map to the same element under the natural quotient

homomorphism Mod(S) → H1(Mod(S); Z). Call this element h. Because Mod(S) is generated by Dehn twists about nonseparating simple closed curves (Corollary 5.15), it follows that H1(Mod(S); Z) is generated by h.

Figure 6.5 A copy of a sphere with four boundary components in a higher genus surface, which gives rise to a lantern relation between 7 nonseparating simple closed curves.

We now claim h is trivial. Since the genus of S is at least 3, it is possible to embed S04 in S so that each of the 7 simple closed curves in S04 involved

The search for the right relation. Mumford was the first to attack the problem of finding the abelianization of Mod(Sg ). He proved that H1(Mod(Sg ); Z) is a quotient of Z/10Z for g ≥ 2 [132]. In his paper, he punctuated his result with a question-exclamation mark, ?!, an annotation used in chess for a “dubious move.” As above, once you kno w that Mod(Sg ) is generated by Dehn twists about nonseparating simple closed curves, it is a matter of using relations between Dehn twists to determine the abelianization. Mumford used the 3–chain relation (TaTbTc)4 = TdTe, hence his result. Birman noticed that one could use a different relation to show that the abelianization of Mod(Sg ) is a quotient of Z/2Z for g ≥ 3 [17, 18]. Powell then produced a product of 15 nonseparating Dehn twists that equals the identity on Mod(Sg ) for g ≥ 3, finally proving Theorem 6.2 [145]. Later, Harer [71] noticed that the lantern relation can be used to give

a simple proof, as above.

For n > 1 the group Mod(Sg,n) is not perfect: if we take the sign of the induced permutation on the punctures (or marked points), we get a surjective homomorphism from Mod(Sg,n) to the abelian group Z/2Z.

THE FIRST HOMOLOGY OF THE MAPPING CLASS GROUP IN LOW GENUS

In order to determine H1(Mod(S); Z) when S is a surface of genus 1 or 2, we work directly from the known presentations of these groups.

Genus two. The group Mod(S2) has the following presentation, due to Birman–Hilden. In the presentation, we use ai to denote the Dehn twist about the simple closed curve ai shown in Figure 6.6.

aiai+1ai = ai+1aiai+1,

(a1a2a3)4 = a25,

[(a5a4a3a2a1a1a2a3a4a5), a1] = 1,

(a5a4a3a2a1a1a2a3a4a5)2 = 1i

The first relation is simply disjointness, the second the bra id relation, and the third a special case of the 3–chain relation (the two simp le closed curves

forming the boundary of the 3–chain are isotopic). The eleme nt a5a4a3a2a1a1a2a3a4a5 appearing in the last two relations is exactly the hyperelliptic involution. We

give the Birman–Hilden proof of this presentation in Chapte r 9, and we give a brief discussion of the “hyperelliptic relations” later i n this section.

To get a presentation for Mod(S2)ab, we simply add the relations that all generators commute. This makes the first and fourth relation s redundant.

be a chain of isotopy classes of simple closed curves in the closed surface Sg ; that is, i(ci, ci+1) = 1 and i(ci, cj ) = 0 when |i −j| > 1. There is only one such chain in Sg up to homeomorphism (this follows from the fact that there is one 2g–chain in Sg up to homeomorphism, as in Section 1.3). The product

Tc2g+1 · · · Tc1 Tc1 · · · Tc2g+1

is a hyperelliptic involution (the hyperelliptic involution when g is equal to 1 or 2).

Thus, we have the following hyperelliptic relations in Mod(Sg ):

(Tc2g+1 · · · Tc1 Tc1 · · · Tc2g+1 )2 = 1 [Tc2g+1 · · · Tc1 Tc1 · · · Tc2g+1 , Tc2g+1 ] = 1

A strange fact. If we rewrite the first hyperelliptic relation, we see that there is a product of 4g + 1 Dehn twists that equals the inverse of one Dehn twist. In other words, a right Dehn twist is a product of left Dehn twists. This, plus the Dehn–Lickorish theorem, gives us the followi ng surprising fact (pointed out to us by Luis Paris):

Every element of Mod(Sg ) is a product of left (positive) Dehn twists.

6.2 PRESENTATIONS FOR THE MAPPING CLASS GROUP

We have already seen several relations between Dehn twists. In particular, we have the disjointness relation (Fact 3.8), the braid relation, the chain relation, the lantern relation, and the hyperelliptic relation. We will see that these relations suffice to give a finite presentation for Mod(Sg ).

WAJNRYB'S PRESENTATION

Finite presentations for the mapping class groups of closed surfaces of genus 1 and 2 were already discussed in Section 6.1. McCool gave the first algorithm for finding a finite presentation for the mapping class g roup of a higher

genus surface [116]. His techniques are algebraic in nature; no explicit presentation has been derived from this algorithm.

Hatcher and Thurston made a breakthrough by finding a topolog ically flavored algorithm for constructing an explicit finite present ation for Mod(S). The algorithm was carried out by Harer, who produced a finite b ut unwieldy presentation [71]. Wajnryb used these ideas to derive the following explicit presentation, which is considered to be the standard presentation for Mod(S) [168, 25]. The exact form of the presentation given here is taken from a survey paper of Birman [21]. In the statement, we use functional notation as usual (elements applied right to left).

surface or a surface with one boundary component and genus g ≥ 3. Let ai denote the Humphries generator Tci , where ci is as shown in Figure 6.7. The mapping class group Mod(S) has a presentation where the generators are a0, . . . , a2g , and the relations are as follows.

where

b0 = (a4a3a2a1a1a2a3a4)a0(a4a3a2a1a1a2a3a4)−1

148 CHAPTER 6

4. Lantern relation

a0b2b1 = a1a3a5b3

where

b1 = (a4a5a3a4)−1a0(a4a5a3a4) b2 = (a2a3a1a2)−1b1(a2a3a1a2)

b3 = (a6a5a4a3a2ua−1 1a−2 1a−3 1a−4 1)a0(a6a5a4a3a2ua−1 1a−2 1a−3 1a−4 1)−1

and where

u= (a6a5)−1b1(a6a5)

5.Hyperelliptic relation (S closed)

(a2g · · · a1a1 · · · a2g )d = d(a2g · · · a1a1 · · · a2g )

where d is any word in the generating set that, under the previous relations, is equivalent to the Dehn twist about the simple closed curve d in Figure 6.8.

b0 d

...

b2 b3

...

b1

Figure 6.8 Extra elements used in the relations for Wajnryb's presentation for Mod(S). We have labelled the simple closed curves by the corresponding elements of

Mod(S).

In the statement, we mean that the hyperelliptic relation is only needed (and it is only true) for closed surfaces. The reason for the term “ hyperelliptic

relation” is that the product d(a2g · · · a1a1 · · · a2g )d is a hyperelliptic involution.

Strictly speaking, Theorem 6.3 does not give a formal presentation of Mod(Sg ), since we have not given the element d in terms of the generators, so we take care of that now. If we rephrase things, we need to write the Dehn twist d as

a product of the generators ai in the mapping class group of the surface with one boundary component. Let n1, . . . , ng be the Dehn twists about the simple closed curves shown in Figure 6.9. Note that n1, n2, and ng are the same as the Dehn twists a1, b0, and d from Theorem 6.3. Similarly to Section 5.4, we can inductively write the ni in terms of the Humphries generators. We start with n1 = a1 and n2 = b0. Then, we have

ni+2 = winiwi−1,

where

wi = (a2i+4a2i+3a2i+2ni+1)(a2i+1a2ia2i+2a2i+1)(a2i+3a2i+2a2i+4a2i+3)(ni+1a2i+2a2i+1a2i).

Finally, set d = ng.

We have labelled the simple closed curves by the corresponding elements of

Mod(S).

A presentation of the mapping class group of a surface with more than one boundary component can be obtained by applying the Birman exact sequence. Also, a presentation for Mod(Sg,1) can be obtained by combining Wajnryb's presentation with Proposition 3.13.

The effect of relations on homology. Harer notes that if we take the abstract group with the Humphries generators and the first two s ets of relations in the Wajnryb presentation, then we have a group (an Artin group) whose first homology is Z. We see from our proof of Theorem 6.2 that if we next add in the lantern relation, the resulting group has trivial first homology. At this point, our abstract group has trivial second homology, yet Harer proved that H2(Mod(Sg ); Z) ≈ Z (Theorem 6.8 below). Adding in the 3–chain relation corrects this.

The algebro-geometric approach. Years before McCool's result, Baily and Deligne–Mumford gave different compactifications of M(S), the moduli space of Riemann surfaces homeomorphic to S, showing that M(S) is a quasiprojective variety [9, 47]. We will prove in Theorem 8.9 below that Mod(S) has a finite index subgroup that is torsion free, from which it follows that M(S) has a finite cover (corresponding to ) which is a manifold, and so a smooth quasiprojective variety. Lojasiewicz had also shown that any smooth quasiprojective variety has the homotopy type of a finite complex; in particular its fundamental group is finitely presen ted. We conclude that , hence Mod(S), is finitely presented. However, this approach does not give an algorithm for finding an explicit finite presentat ion.

THE CUT SYSTEM COMPLEX

We now very briefly outline the strategy used to derive the pre sentation in Theorem 6.3. In Section 6.3 below, we will give a complete proof that Mod(Sg ) is finitely presented, although we will not derive an explici t presentation.

Hatcher–Thurston [74] defined a 2–dimensional CW–complex X(Sg ), called the cut system complex, as follows. Vertices of X(Sg ) correspond to cut systems in Sg , that is (unordered) sets {c1, . . . , cg } where:

1.each ci is the isotopy class of a nonseparating simple closed curve γi in Sg ,

2.i(ci, cj ) = 0 for all i and j, and

3.Sg − γi is connected.

An example of a vertex in X(Sg ) is given by the set of isotopy classes {a1, . . . , ag } shown in Figure 6.10. Vertices represented by {ai} and {bi} are connected by an edge in X(Sg ) if (up to renumbering) ai = bi for

2 ≤ i ≤ g, and i(a1, b1) = 1.

Just as the edges of X(Sg ) are defined by certain topological configurations of curves, so are the 2–cells of X(Sg ). For example, we glue in a triangle to the 1–skeleton of X(Sg ) for each triple of vertices that are pairwise connected by edges. For example, in Figure 6.10, the vertices va = {a, a2, . . . , ag }, vb = {b, a2, . . . , ag }, and vc = {c, a2, . . . , ag } span a triangle in X(Sg ). The complex X(Sg ) also has squares and pentagons; we refer the reader to the paper [74] for the details.